Republicans are urging Bush, the president's younger brother, to strongly consider making a bid for the seat currently held by outgoing Republican Sen. Mel Martinez.

The first-term senator announced Tuesday he will not run for re-election, setting off what's likely to be one of the most hotly contested Senate battles in 2010.

Bush, who served as Florida governor from 1999 until 2007 and enjoyed high approval ratings, will take the next month to consider getting into the race, Castellanos, a CNN contributor, said on CNN's "The Situation Room" on Tuesday.

Bush also told Politico.com he is "considering" getting into the race.

Should Bush decide to run, his presence would add immediate star power to a race Democrats are likely to make their top target in 2010. But the former governor, who has largely remained out of the spotlight over the past two years, could be hindered by his close association with the unpopular outgoing president.

"Let me give him some free advice," said Paul Begala, a CNN contributor. "Change your name. Run as John Ellis, not John Ellis Bush. The Bush brand is probably what croaked Mel Martinez."

Don't Miss

According to Florida exit polls on Election Day, just 9 percent of Floridians approve of President Bush's job performance. But Jeb Bush left office in 2007 with approval ratings well over 60 percent, a sign he could be insulated in the state from his brother's unpopularity.

Since exiting the governor's office, Bush has served on the board of Tenet Healthcare Corporation. He also joined the now-defunct investment bank Lehman Brothers in 2007 as an adviser to its private equity business.

Bush endorsed Sen. John McCain's presidential bid last February though rarely campaigned on his behalf.

According to published reports, Democrats view Alex Sink, the state's chief financial officer, as the candidate best positioned to run for the seat. Before Martinez's announcement, Sink suggested she was uninterested in the run, but it's likely she now may reconsider.

Democrats are also considering Rep. Debbie Wasserman Schultz, a high-profile supporter of Sen. Hillary Clinton's presidential bid, though Schultz's congressional spokesman said it is unlikely she will run.

Other Democrats mentioned include Reps. Kendrick Meek and Allen Boyd.

On the Republican side, Reps. Vern Buchanan and Connie Mack are said to be interested, as well as several GOP state officials.